Timeline for How to get 3-manifold, Knots from Number Fields
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 28, 2013 at 0:04 | comment | added | Emerton | @JohnMangual: P.P.S. See in particular the second para. of the introduction (so the second para. on p1) for a statement by the authors along the lines of my comments. (Note that inner forms means e.g. passing between arithmetic subgroups of $GL_2(K)$ and arithmetic subgroups of a quaternion algebra over $K$, where $K$ is a quad. imaginary field.) | |
| Jun 27, 2013 at 23:58 | comment | added | Emerton | @JohnMangual: P.S. Perhaps I should add that the JL correspondence is about relationships between automorphic forms (which are harmonic representatives for cohomology classes with $\mathbb C$-coeffs.) between various different arithmetic hyperbolic 3-manifolds, and a major goal of the CV book is to extend this correspondence to torsion classes (to which traditional Hodge theory, and hence the theory of automorphic forms, don't apply). There are various reasons for suspecting that JL carries over to this context, the anticipated relationship with Galois representations being a primary one. | |
| Jun 27, 2013 at 23:55 | comment | added | Emerton | @JohnMangual: Dear John, The book of CV is not about a relationship between knots/links and number theory. It is about the relationship between (certain, namely congruence) arithmetic hyperbolic 3-manifolds and number theory. In particular, it is about the relationship (mainly conjectural at the time the book was written, though less so now thanks to recent work of Calegari--Geraghty and Peter Scholze) between torsion classes in the (co)homology of these manifolds and Galois representations. (This is basically what FJ writes in the last paragraph of their answer.) Regards, | |
| Jun 27, 2013 at 23:29 | comment | added | john mangual | Ms Jellyfish, I have unchecked this answer to see if we can get a better response. | |
| May 15, 2013 at 14:19 | comment | added | Neil Hoffman | To continue Black's point about the "over"-representation of arithmetic manifolds, there are only finitely many arithmetic manifolds below a given volume, so the density of arithmetic manifolds among all manifolds volume below that of the Whitehead link will be 0. Also, Neumann and Reid's paper "Arithmetic of Hyperbolic manifolds" is a good resource if you are looking for a quicker (though obviously not as thorough) treatment of a number of these ideas. | |
| May 15, 2013 at 14:00 | vote | accept | john mangual | ||
| Jun 27, 2013 at 23:29 | |||||
| May 15, 2013 at 1:39 | history | answered | user631 | CC BY-SA 3.0 |